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At a picnic there were 3 times as many adults as children and twice as many women as men. If there was a total of \(\mathrm{x}\) men, women, and children at the picnic, how many men were there, in terms of \(\mathrm{x}\)?
Let's break down what each phrase means in everyday language:
"3 times as many adults as children" means: If there are some number of children, then the number of adults is 3 times that amount. For example, if there are 5 children, there would be 15 adults.
"Twice as many women as men" means: If there are some number of men, then the number of women is 2 times that amount. For example, if there are 10 men, there would be 20 women.
We also know that adults consist of both men and women (no other categories), and the total of everyone (men + women + children) equals x.
Our goal is to find how many men there were, expressed as some fraction of x.
Process Skill: TRANSLATE - Converting the English relationships into mathematical understanding
Since we want to find the number of men, let's use that as our starting point. We'll call the number of men "M".
From our translation:
Now, since there are "3 times as many adults as children":
This gives us:
The total number of people at the picnic is everyone added together:
\(\mathrm{Total = Men + Women + Children}\)
Substituting our expressions:
\(\mathrm{Total = M + 2M + M = 4M}\)
Since we're told the total equals x:
\(\mathrm{4M = x}\)
We have the equation: \(\mathrm{4M = x}\)
To find M (the number of men) in terms of x, we divide both sides by 4:
\(\mathrm{M = x/4}\)
Let's verify this makes sense:
Also checking our original relationships:
The number of men at the picnic is \(\mathrm{x/4}\).
This corresponds to answer choice C.
1. Misinterpreting "3 times as many adults as children"
Students often confuse the direction of this relationship. They might think it means "\(\mathrm{adults = children ÷ 3}\)" instead of "\(\mathrm{adults = 3 \times children}\)". This fundamental misunderstanding of the multiplier relationship will lead to completely incorrect variable setup.
2. Forgetting that adults = men + women
Students may treat men, women, adults, and children as four separate, independent groups instead of recognizing that adults is composed of men and women. This leads to setting up equations with four variables instead of understanding the hierarchical relationship.
3. Starting with the wrong base variable
Since the question asks for men in terms of x, students might try to start by defining children or women as their base variable, making the algebra unnecessarily complex and increasing the chance of errors in the relationship setup.
1. Algebraic errors when expressing children in terms of men
When students have "\(\mathrm{adults = 3M}\)" and "\(\mathrm{adults = 3 \times children}\)", they may incorrectly solve for children, getting "\(\mathrm{children = 3M}\)" instead of "\(\mathrm{children = M}\)". This happens when they don't properly cancel the coefficient of 3 on both sides.
2. Arithmetic mistakes when adding fractions in verification
When checking their work with \(\mathrm{x/4 + x/2 + x/4 = x}\), students often struggle with adding fractions, particularly converting \(\mathrm{x/2}\) to \(\mathrm{2x/4}\), leading them to doubt their correct answer.
3. Setting up the total equation incorrectly
Students may write the total as "\(\mathrm{3M + M = x}\)" (thinking adults + children) instead of "\(\mathrm{M + 2M + M = x}\)" (men + women + children), forgetting to break down adults into its constituent parts.
1. Solving for the wrong quantity
After finding \(\mathrm{M = x/4}\), students might accidentally select the answer choice corresponding to women (\(\mathrm{x/2}\)) or adults (\(\mathrm{3x/4}\)) instead of men, especially if they calculated these quantities during their verification process.
2. Inverting the final fraction
From the equation \(\mathrm{4M = x}\), students might incorrectly solve to get \(\mathrm{M = 4x}\) instead of \(\mathrm{M = x/4}\), then try to match 4x to one of the given fractional answer choices, leading to confusion.
Step 1: Choose a strategic smart number for the total
Since we need to work with relationships involving multiples of 2 and 3, let's choose \(\mathrm{x = 12}\) people total. This number works well because it's divisible by the factors we'll encounter.
Step 2: Set up the relationships with our smart number
Let M = number of men
Then: \(\mathrm{Women = 2M}\) (twice as many women as men)
\(\mathrm{Children = C}\)
\(\mathrm{Adults = M + 2M = 3M}\)
Since \(\mathrm{adults = 3 \times children: 3M = 3C}\), so \(\mathrm{C = M}\)
Step 3: Express total in terms of M
\(\mathrm{Total = Men + Women + Children}\)
\(\mathrm{12 = M + 2M + M = 4M}\)
Therefore: \(\mathrm{M = 12 ÷ 4 = 3}\) men
Step 4: Verify our relationships work
Step 5: Find the pattern
With \(\mathrm{x = 12}\), we have 3 men.
The ratio is \(\mathrm{3/12 = 1/4}\)
Therefore: Number of men = \(\mathrm{x/4}\)
Answer: C. \(\mathrm{x/4}\)